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I am teaching myself mathematics, my objective being a thorough understanding of game theory and probability. In particular, I want to be able to go through A Course in Game Theory by Osborne and Probability Theory by Jaynes.

I understand I want to cover a lot of ground so I'm not expecting to learn it in less than a year or maybe even two. Still I'm fairly certain it's not impossible.

However I would like to have a study plan more or less fleshed out just to know I'm on the right track. There were some other questions related to self learning math here but I couldn't find one like mine.

I'd appreciate some feedback.

Calc I + II: no book, I already know basic calculus

  • Differential equations: MIT's OCW lectures
  • Calc III: Stewart's Multivariable calculus
  • Linear Algebra: Strang, Gilbert, Linear Algebra and Its Applications complemented with MIT's OCW lectures OR Linear Algebra Done Right

Until here I am more or less certain on what I want to study, but I'm totally confused on what to learn next. Jayne's book states that you need to be familiar with applied mathematics.

After reading about applied mathematics, I came up with this plan to be done after finishing what I mentioned earlier (in order of course, not all at the same time):

  1. Topology A: Munkres, part I.
  2. Real analysis: Still not sure about the material, probably Abbott or Rudin.
  3. Complex Analysis: No idea about the material
  4. Group Theory: Rotman, An Introduction to the Theory of Groups
  5. Topology B: Munkres, part II.

And then finally, Jayne's Probability Theory and game theory.

Am I missing something here? Some of these books such as Rotman's are aimed at a graduate level, is it foolish to think I will understand them?

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    $\begingroup$ To get a good understanding of probability theory you need a background on measure theory. $\endgroup$ – Integral Oct 1 '14 at 20:34
  • $\begingroup$ Thank you, that's exactly the kind of feedback I was looking for. $\endgroup$ – ignis Oct 1 '14 at 21:02
  • $\begingroup$ To add to @Integral : I'd put it right after Real Analysis. $\endgroup$ – user76844 Oct 2 '14 at 3:11
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Perhaps this should be a comment, but I don't have the reputation. I TA a class that uses the Osborne and Rubinstein text you mention. I can't speak in depth regarding your goals in probability, but here's my advice on Game Theory:

Unless you're looking to publish research in game theory, the mathematics required is not that substantial. You already have a background in Calculus. On top of that, I would do a little work in analysis (I recommend Rudin) and maybe some optimization - Mathematics for Economists by Simon and Blume is a good reference for that. If you're doing more advanced problems, then you'll want to get good at proofs, particularly direct, by contradiction and by induction. Again, Rudin's good for that. Analysis is hard work - if this is your first time doing it, don't be intimidated. Just slog through it! You could take some time on fixed point and separating hyperplane theorems, but you'll rarely use them.

Once you have that under your belt, you're more than ready to start game theory. I might recommend you start with a slightly easier game theory book and then move up to A Course in Game Theory. My favorite undergrad GT textbook is Strategy by Watson. But if you like Osborne, then he also has an introductory text called An Introduction to Game Theory. I'd start with skimming through one those texts before tackling A Course in Game Theory, which goes a little deeper and provides a little less explanation.

Sometimes students come to me and say that they're struggling because they can't do the math required in game theory. Very rarely is this actually the case. The real issue is that game theory is notationally very dense and the professor I TA for values formalism (as do I). For students who have not done a lot of math, this notational density really intimidates them. Anyway - don't let anybody tell you that Game Theory is all about difficult math - it isn't. It just look like it is!

By the way, I think it's admirable what you're doing and I wish you all the best in your self-learning.

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I'll second Shane's comments, but I think you should start studying game theory now. Schelling's The Strategy of Conflict is a classic, and very readable with very little math. A more formal book that I've read is Vorob'ev Game Theory (springer-verlag), about the level of an undergraduate math book. An undergraduate game theory book will give you a taste of what is going on, and then you can decide if you want to dig in deeper.

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If you are more or less new to mathematics, your priority might be to train proof skills (by doing lost of easy proofs) and gain comfort with basic properties of sets and functions for example (Introduction to Metric and Topological Spaces by Sutherland gives a good quick overview within an advanced framework, I think).

Such grounding work sounds necessary to me if you want to proactively read mathematical statements in the books you mention in the short-run.

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